Axisymmetric Flows with Swirl for Euler and Navier–Stokes Equations

IF 2.6 2区 数学 Q1 MATHEMATICS, APPLIED
Theodoros Katsaounis, Ioanna Mousikou, Athanasios E. Tzavaras
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Abstract

We consider the incompressible axisymmetric Navier–Stokes equations with swirl as an idealized model for tornado-like flows. Assuming an infinite vortex line which interacts with a boundary surface resembles the tornado core, we look for stationary self-similar solutions of the axisymmetric Euler and axisymmetric Navier–Stokes equations. We are particularly interested in the connection of the two problems in the zero-viscosity limit. First, we construct a class of explicit stationary self-similar solutions for the axisymmetric Euler equations. Second, we consider the possibility of discontinuous solutions and prove that there do not exist self-similar stationary Euler solutions with slip discontinuity. This nonexistence result is extended to a class of flows where there is mass input or mass loss through the vortex core. Third, we consider solutions of the Euler equations as zero-viscosity limits of solutions to Navier–Stokes. Using techniques from the theory of Riemann problems for conservation laws, we prove that, under certain assumptions, stationary self-similar solutions of the axisymmetric Navier–Stokes equations converge to stationary self-similar solutions of the axisymmetric Euler equations as \(\nu \rightarrow 0\). This allows to characterize the type of Euler solutions that arise via viscosity limits.

Abstract Image

欧拉方程和纳维-斯托克斯方程中带有漩涡的轴对称流动
我们将带有漩涡的不可压缩轴对称纳维-斯托克斯方程视为龙卷风样流的理想化模型。假设与边界表面相互作用的无限漩涡线与龙卷风核心相似,我们寻找轴对称欧拉方程和轴对称纳维-斯托克斯方程的静止自相似解。我们尤其关注这两个问题在零粘度极限下的联系。首先,我们为轴对称欧拉方程构建了一类显式静止自相似解。其次,我们考虑了不连续解的可能性,并证明不存在滑移不连续的自相似静止欧拉解。这一不存在的结果被扩展到一类通过涡核有质量输入或质量损失的流动。第三,我们将欧拉方程的解视为纳维-斯托克斯解的零粘度极限。利用守恒定律黎曼问题理论的技术,我们证明了在某些假设条件下,轴对称纳维-斯托克斯方程的静态自相似解收敛于轴对称欧拉方程的静态自相似解(\nu \rightarrow 0\ )。这样就可以确定通过粘度极限产生的欧拉解的类型。
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来源期刊
CiteScore
5.00
自引率
3.30%
发文量
87
审稿时长
4.5 months
期刊介绍: The mission of the Journal of Nonlinear Science is to publish papers that augment the fundamental ways we describe, model, and predict nonlinear phenomena. Papers should make an original contribution to at least one technical area and should in addition illuminate issues beyond that area''s boundaries. Even excellent papers in a narrow field of interest are not appropriate for the journal. Papers can be oriented toward theory, experimentation, algorithms, numerical simulations, or applications as long as the work is creative and sound. Excessively theoretical work in which the application to natural phenomena is not apparent (at least through similar techniques) or in which the development of fundamental methodologies is not present is probably not appropriate. In turn, papers oriented toward experimentation, numerical simulations, or applications must not simply report results without an indication of what a theoretical explanation might be. All papers should be submitted in English and must meet common standards of usage and grammar. In addition, because ours is a multidisciplinary subject, at minimum the introduction to the paper should be readable to a broad range of scientists and not only to specialists in the subject area. The scientific importance of the paper and its conclusions should be made clear in the introduction-this means that not only should the problem you study be presented, but its historical background, its relevance to science and technology, the specific phenomena it can be used to describe or investigate, and the outstanding open issues related to it should be explained. Failure to achieve this could disqualify the paper.
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